A Polynomial Analogue of Landau's Theorem and Related Problems

Gorodetsky, Ofir

doi:10.1112/s0025579317000092

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…Secondly, O K is real quadratic and has narrow class number one. For the cases D = −1 and D = −t, a polynomial analogue of Landau-Shanks' problem has been studied by Bary-Soroker, Smilansky, Wolf [1] and Gorodetsky [5]. We generalize their results to square-free polynomials D ∈ A and prove an analogue of Landau-Shanks' theorem as follows.…”

Section: Introductionmentioning

confidence: 93%

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Variation of a Theme of Landau--Shanks in Positive Characteristic

Chuang¹,

Kuan²,

Yao³

2021

Taiwanese J. Math.

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Let A := F q [t] be a polynomial ring over a finite field F q of odd characteristic and let D ∈ A be a square-free polynomial. Denote by N D (n, q) the number of polynomials f in A of degree n which may be represented in the form u•f = A 2 −DB 2 for some A, B ∈ A and u ∈ F × q , and by B D (n, q) the number of polynomials in A of degree n which can be represented by a primitive quadratic form of a given discriminant D ∈ A, not necessary square-free. If the class number of the maximal order of F q (t, √ D) is one, then we give very precise asymptotic formulas for N D (n, q). Moreover, we also give very precise asymptotic formulas for B D (n, q).

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Section: Introductionmentioning

confidence: 93%

“…Using Theorem 1.5, we can get asymptotic formulas, as n tends to ∞, in all theorems of this paper. In [5], Gorodetsky refine Theorem 1.5 to Theorem 1.6 and use Theorem 1.6 deduce the asymptotic formula as q n tends to ∞ for D = −t.…”

Section: (C) Whenmentioning

confidence: 98%

Variation of a Theme of Landau--Shanks in Positive Characteristic

Chuang¹,

Kuan²,

Yao³

2021

Taiwanese J. Math.

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“…There are several ways that q n can tend to infinity. The above is valid when q is much larger than n. For all ways that q n can tend to infinity, Gorodetsky [3] proved that…”

Section: Introductionmentioning

confidence: 94%

“…which is the same as the i-th coefficient of (3) above. Now, the matrix in (3) is such that all entries on a given skew-diagonal are identical, and thus it is a Hankel matrix. This is similar to a Toeplitz matrix, where the entries on a given diagonal are identical.…”

Section: Motivationmentioning

confidence: 99%

The Variance of the Sum of Two Squares over Intervals in $\mathbb{F}_q [T]$: I

Yiasemides¹

2022

Preprint

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For B ∈ F q [T ] of degree 2n ≥ 2, consider the number of ways of writing B = E 2 + γF 2 , where γ ∈ F * q is fixed, and E, F ∈ F q [T ] with deg E = n and deg F = m < n. We denote this by S γ;m (B). We obtain an exact formula for the variance of S γ;m (B) over intervals in F q [T ]. We use the method of additive characters and Hankel matrices that the author previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach; and we briefly discuss the possible extension of our result to the number of ways of writing B = E 2 + T F 2 .

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“…Unless stated otherwise, constants, both implied and explicit, are absolute. For recent results where new function field estimates are obtained by comparing to a permutation quantity see [Gor17,EG20].…”

Section: Introductionmentioning

confidence: 99%

Uniform estimates for friable polynomials over finite fields

Gorodetsky¹

2022

Preprint

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We establish new estimates for the number of m-friable polynomials of degree n over a finite field Fq, where the main term involves the number of m-friable permutations on n elements. The range of m in our results is n ≥ m ≥ (1 + ε) log q n and is optimal as these numbers are not comparable for smaller m. For m ≥ 4 log q n the error term in our estimates decays faster than in all previous works.Our estimates imply that the probability that a random polynomial fn is m-friable is asymptotic to the probability that a random permutation πn is m-friable, uniformly for m ≥ (2 + ε) log q n as q n → ∞. This should be viewed as an unconditional analogue of a result of Hildebrand (1984), saying that under the Riemann Hypothesis, the probability that a number ≤ x is y-friable is of the order of magnitude of the Dickman ρ function at log x/ log y, uniformly for log y ≥ (2 + ε) log log x. As opposed to Hildebrand's result, we actually obtain an asymptotic result.As an application of our estimates, we determine the rate of decay in the asymptotic formula for the expected degree of the largest prime factor of a random polynomial. This improves a previous estimate of Knopfmakher and Manstavičius (1997).

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A Polynomial Analogue of Landau's Theorem and Related Problems

Cited by 7 publications

References 21 publications

Variation of a Theme of Landau--Shanks in Positive Characteristic

Variation of a Theme of Landau--Shanks in Positive Characteristic

The Variance of the Sum of Two Squares over Intervals in $\mathbb{F}_q [T]$: I

Uniform estimates for friable polynomials over finite fields

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